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Akisation here is applied to the hexagonal faces of tut only. Regular triangles have sides xxx, acute triangles have sides xzz.
Height of those pyramids is to be chosen as to match the same circumradius. From this height setting it follows that z = sqrt[11-sqrt(33)]/2 = 1.146237.

Incidence matrix according to Dynkin symbol

xo3xo3oy&#z   where: y = sqrt(11/3) = 1.914854 (tip distance, pseudo edge)
                     z = sqrt[11-sqrt(33)]/2 = 1.146237

o.3o.3o.    | 12 * | 1  2  2 | 1  2  2
.o3.o3.o    |  * 4 | 0  0  6 | 0  3  3
------------+------+---------+--------
x. .. ..    |  2 0 | 6  *  * | 0  2  0  x
.. x. ..    |  2 0 | * 12  * | 1  0  1  x
oo3oo3oo&#z |  1 1 | *  * 24 | 0  1  1  z
------------+------+---------+--------
.. x.3o.    |  3 0 | 0  3  0 | 4  *  *  xxx
xo .. ..&#z |  2 1 | 1  0  2 | * 12  *  xzz
.. xo ..&#z |  2 1 | 0  1  2 | *  * 12  xzz